If \(\pi = \mathcal {N}({\bf 0},\Sigma )\) , then describing the sparsity of a linear KR
rearrangement due to conditional independence reduces to the problem
of characterizing the fill-in for the Cholesky factor of the precision matrix \(\Sigma ^{-1}\) —a well-known result
in the field of Gaussian graphical models
[1]}, which has also been exploited in the context of data
assimilation to devise regularized estimators of the ensemble covariance
[2]}.
Here, we consider the
non-Gaussian generalization provided by
[3]}, where the Cholesky factor is replaced by a
nonlinear KR rearrangement.
In particular, we note that
the sparsity of \(S\)  depends on the ordering of the input variables in the same way as the fill-in of the Cholesky factor does. Different orderings can yield different sparsity patterns.
See Figure REF  (right) for an example.
Whenever possible,
we then look for orderings that promote sparsity
(cf. [3]} for
additional details).
